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algorithms are referred to as SVM ndcg and SVM mrr . Basically, in these extensions,
different feature maps or different strategies of searching the most violated con-
straints are used, but the key idea remains the same as that of SVM map .
Further analysis in [ 33 ] shows that the loss functions in the aforementioned works
are actually convex upper bounds of the following quantity, and this quantity is an
upper bound of the corresponding measure-based ranking error.
y 1
M y c , y I
max
y c
,
(4.19)
w T Ψ( y , x )
w T Ψ( y c , x )
{
}
=
where M( y c , y ) represents the value of an evaluation measure M when the ranking
result is y c and the ground truth label is y .
In [ 5 , 6 , 19 , 36 ] the following convex upper bound of the above quantity is mini-
mized:
y 1
M y c , y +
w T Ψ y c , x
w T Ψ( y , x )
+
max
y c
.
(4.20)
=
In another method, called PermuRank [ 33 ], a different convex upper bound of the
aforementioned quantity is employed in the optimization process, as shown below.
y 1
M y c , y 1
w T Ψ y c , x
w T Ψ( y , x )
max
y c
+
.
(4.21)
+
=
4.2.3 Non-smooth Optimization
Different from the methods introduced in the previous two subsections, there are
also some other methods that directly optimize evaluation measures using non-
smooth optimization techniques. For example, in AdaRank [ 32 ], the boosting algo-
rithm is used to optimize the exponential function of the evaluation measure. Note
that since the exponential function is monotonic, the optimization of the objective
function in AdaRank is equivalent to the optimization of the evaluation measure it-
self. For another example, in RankGP [ 34 ], the evaluation measure is used as the
fitness function of a genetic algorithm.
It should be noted that although these non-smooth optimization techniques are
very general and can deal with the optimization of any non-smooth functions, they
also have their limitations. That is, although the evaluation measures are used di-
rectly as the objective function, it is not guaranteed that one can really find their
optima.